On structural topology optimization using graded porosity control

  • Martin-Pierre SchmidtEmail author
  • Claus B. W. Pedersen
  • Christian Gout
Research Paper


In recent years, the field of additive manufacturing (AM), often referred to as 3D printing, has seen tremendous growth and radically changed the way we describe valid 3D models for fabrication. While not free of constraints, AM offers an unprecedented level of freedom in geometrical complexity for manufacturable feasible designs. One example of such design freedom is the creation of intricate, robust, and lightweight internal structures. Our approach builds upon and extends the recent works on topology optimization for the so-called infill structures. In order to have more design control over these infill structures, we present a new formulation allowing the generation of mixed design patterns containing bulk and porous regions using a guiding constraint parameterized by non-uniform constraining fields. Secondly, we demonstrate multiple methods of generating such non-uniform fields to leverage the present formulation and analyze their effect on the geometrical and physical properties of the obtained designs.


Topology optimization Finite element analysis Mathematical programming Local volume constraints Guided porosity control Additive manufacturing 



The authors are thankful to the reviewers for their insightful comments helping improve the paper and to the members of the Department of Mechanical Engineering and Department of Electrical Engineering of the Technical University of Denmark for the organization of the 2017 PhD course “Topology Optimization - Theory, Methods and Applications.” C. Gout thanks M2SiNum project (co-financed by the European Union and by the Normandie Regional Council) and CIEMME OpenMod platform (INSA Rouen) for their support.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

Replication of results

Unless explicitly stated, all optimized designs of the present paper used the following parameters: γ = 3, Y 0 = 1Ymin = 10− 6, \(G_{l}^{\star }= 0.6\), Poisson ratio of 0.3, isotropic local neighborhood of radius rE = 4, a sensitivity filter of radius 1.3 with elements of size 1 arranged in a regular 2D or 3D grid of squares or cubes, respectively.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Martin-Pierre Schmidt
    • 1
    Email author
  • Claus B. W. Pedersen
    • 2
  • Christian Gout
    • 1
  1. 1.LMIINSA Rouen NormandieSaint-Étienne-du-RouvrayFrance
  2. 2.HamburgGermany

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